We have been developing a new MRI modality that we call Diffusion Tensor MRI. It consists of a) estimating an effective diffusion tensor, D, in a voxel from a series of diffusion-weighted images, and b) using D to derive useful information about tissue structure and function. We produce images of diffusion ellipsoids that depict local fiber orientation and mean diffusion distances, as well as images of scalar invariants of D that are independent of the reference frame in which magnetic field gradients and D are measured. Quantitative Diffusion Tensor MRI is what we call the calculation and display of MR imaging parameters that behave like histological or physiological stains. These include "stains" for diffusion isotropy, diffusion anisotropy, structural similarity, and fiber-tract organization. To account for the complex interactions between imaging and diffusion gradient pulses applied during the acquisition of diffusion-weighted images, we have derived analytic expressions of the "b-matrix" for general 2-D FT and EPI diffusion-weighted sequences. These b-matrices are calculated for each diffusion-weighted image in order to estimate D off-line. More recently, we have developed b-matrices for ultra-fast sequences (e.g., interleaved EPI) that allow Diffusion Tensor Imaging to be performed at a sufficiently high resolution and speed for radiological assessment in a clinical setting.